Evaluation using statistical analysis of different policies for the inspections of the catenary contact wire

Transcription

1 Evaluation using statistical analysis of different policies for the inspections of the catenary contact wire 1&2 J. Casaert, 1 F. Sourget, 1 R. Ziani, 2 D. Bosq, 1 M. Antoni SNCF, Paris, France 1 ; Université Pierre et Marie Curie, Paris, France 2 Abstract In this article, we will present some statistical models built to provide answer to a preventive maintenance problematic: evaluation of the different policies for the inspections of the catenary contact wire. This problematic is an important issue for the SNCF, the French national Railway Company, who searches to increase maintenance efficiency while controlling the costs. The research on this topic is divided into two parts: a spatial modeling in order to decide how frequent the measures along the wire must be and a temporal modeling to check the inspection periodicities of the contact wire. 1. Introduction For any infrastructure manager, maintenance of the catenary contact wire, like maintenance of other components, is a very important issue. To increase maintenance efficiency while controlling the costs, new technologies are introduced. In this context, new inspections vehicles have been used since 2000 at SNCF. These vehicles enable to measure automatically the contact wire width while running at low speed. The introduction of these vehicles has changed the way the maintenance of the contact wire is performed at SNCF. Therefore, it is necessary to evaluate the impact and discrepancies inherent to this new policy of maintenance. In this article, we will first present the new inspection vehicles and the technology they rely on, followed by a brief description of the difference between the old and new policies. Next, we will focus on the statistical analysis carried out on the feedback data. This analysis aims at achieving two different goals. The first goal is to model the degradation of the contact wire along the rail track and to characterize the spatial structure of its degradation. The second goal is to model the temporal degradation of contact wire. Results of this analysis will be presented. Finally, prospective studies will be discussed based on the outputs of the present work. 2. Inspections policies and measurement systems Until 2000, inspections of the catenary contact wire were based on manual measurements realised on every support and at every mid-span between two supports. These measurements were realised by catenary specialists who were taking advantage of the intervention in order to visually check the whole catenary system. Since 2000, two new measurement vehicles have been introduced: CYBERNETIX and MEDES. - CYBERNETIX allows measuring the contact wire thickness every two centimetres while running at 5 km/h (maximal speed is 30km/h, but limited to 5km/h to allow a visual inspections of the catenary defects at the same time). The measurement is based on an optical system and is very sensitive to the location of connectors. The system measures the wire thickness under voltage and uses a specific pantograph which includes a laser micrometer (performing the measures) and an IR support detector both. CYBERNETIX can measure one or two wires at the same time. - MEDES performs measurements of the wearing flat every 20 cm while running at 120 km/h. So MEDES also realizes measurements under voltage but, unlike CYBERNETIX, without contact. The measures are based on the reflection of a laser beam on the flat of contact wire registered on LCD cameras. MEDES even allows getting measurements in the zones where there are more than one wire (until four wires) as well as in connectors area. SNCF has introduced these new measurement vehicles in order to:

2 - increase the number and accuracy of the measurements - improve its capacity to build a significant feedback on wire evolution - decrease the inspection costs, mainly impacted by manual measurement - increase the availability of the lines - improve quality of the feedback database The vehicle supporting MEDES is used at the beginning of the contact wire life. As soon as MEDES measurement is not sufficient (two many errors of measurement), CYBERNETIX is used on the sector. In the same way, if the results obtained by CYBERNETIX inspections are not good enough, some manual measurements can be made to get more information on the considered sector of the contact wire. On one hand, we would like to ensure that the criterions to determine if MEDES is satisfactory are well defined and if the use of MEDES is justified. On the other hand, we would like to know if the periodicities of inspections are good (not too short neither too large). 3. Statistical Analysis Answer to these two questions is based on statistical analysis. Two different models are requested: - spatial model enabling to determine as well as possible the criterion to use CYBERNETIX instead of MEDES - temporal model enabling to optimise the periodicities of inspections In a first subpart of this paper, we will discuss the catenary classification to be built to develop representative models of the network catenary. Then, one subpart for each dimension (spatial and time) will be developed. Above all, one must precise that two main catenaries are used on the French railway network: - contact wire under V - contact wire under 1 500V The degradation of the 1500V catenary is bigger than the V one, and concerns a larger part of the railway network. For these two reasons, SNCF experts decided to focus their attention on the degradation of the 1 500V catenary Catenary network classification If we want to develop representative models of the catenary network, we first need to have an idea of the different existing types of catenary: indeed, we have to distinguish the catenaries which follow different speeds of degradation, otherwise the models would not be consistent. We would like to work on homogenous clusters of catenary with regards to life span. However, the expected lifetime of the catenary is not known yet, so we must begin by looking for some variables which can both impact the lifetime and are available for every section. The method used to find variables related to lifetime is a decision tree. Decision tree A decision tree is a classical supervised classification learning method. It separates observations represented by one variable (noted Y) into different groups. The observations of each group have as much as possible similar characteristics concerning Y and the different groups are the more distinct as possible relatively to Y. Variables which enable to distinguish the different groups are those which have more influence on the behaviour of Y. [1] Here is the general outline of the analysis done:

3 Figure 1: outline of the classification analysis The initial data used is a table in which are recorded all contact wires which have been replaced since For each section of replaced wire are given different kinds of variables: - traffic information - material elements composing the catenary - track profile - the lifetime of the contact wire Before beginning the analysis, catenary have been distinguished (all concerning only the 1 500V catenary) following its geometry: round or méplat. Then, decision tree have been computed for the two groups. Actually, it was already known that the performance of the contact wire is not the same for the two characters. At the moment, the analysis has been already carried out two times (in accordance with the sketch above). Every time, the characterization founded was not sufficient to explain all the heterogeneity of the network. To overcome this problem, experts suggested adding some variables (like: number of pantographs pro year running on the sector, type of the bande de frottement, type of the locomotive). In fact, these variables were not known in the initial data but have certainly an influence on the lifespan of the contact wire. Moreover, the discussions with experts also lead to study more precisely the correlations between the different variables unless some effect can be hidden by other. This stage is still running, definitive results are not yet available, but some main effects are already detected. In fact, the following variables have a big influence on the behaviour of the contact wire: - UIC class (indicator of the line traffic) - mean running speed of the sector (on which the contact wire is attached) - composition of the contact wire (copper, aluminium ) - number of pantographs running on the sector - engine type 3.2. Spatial modeling Spatial model must enable to calculate the likelihood for a critical point not to be detected while measurements are taken every 20 cm (MEDES case). To model degradation in space, measurements very close to each other are necessary. We use data measured with CYBERNETIX, that is, measurements of the thickness every 2 cm for different kinds of section wire. CYBERNETIX society (which provides the system) announces a 0.05 mm accuracy for the wire section measurement, and output data from the system are rounded to mm. SNCF s experts estimate that system accuracy is rather in the order of 0.1mm in the real conditions of use. By

4 considering these elements and in order to improve the models, we choose to work with measures rounded to 0.01 mm. Geostatistics is the method proposed to find a spatial model. After having carried out the variogram study on the different classes of points (closer or not to the connectors), ordinary krigging will be used to fit the data. This approach has the advantage to be specific to spatial fitting and to include all the surrounding neighbours of a point of interest. Variogram Variogram is a function used in geostatistical theory to represent the spatial structure of a variable. Note Z a spatial variable evolving on D. In our case, Z(x) is the thickness of the contact wire measured at the position x and D the contact wire itself. Note h a distance between two points of the wire. We put as hypothesis that (Z(x), x in D) is a weak stationary process: its mean and variance do not change over position (E [Z(x)] = m for all position x and Var (Z(x)) = v for all position x). In that case, the variogram (noted G) just depends on the autocovariance C. Recall that the autocovariance of the variable Z at the position h is: C (h) = E [(Z(x)-m) (Z(x+h)-m)] The variogram at the position h is: And we have the followed property: G (h) = C (0) C (h) G (h) 2 C (0) An important concept of the variogram is the range. If the variogram reaches a limited value (called the threshold), it means that there is a distance beyond which Z(x) and Z(x+h) are uncorrelated. This distance is the range. [2] In case of stationary process, the range always exists because of the above property. Kriging Ordinary kriging is an interpolation method which is based on the variogram function. This method has been developed by Georges Matheron in the 60 s. [3] If we want to interpolate the thickness at the position x 0 thanks to krigging, we use the expression: Z* (x 0 ) = Σ i λ i (x 0 ) Z(x i ) for i in D (x 0 ) Where D (x 0 ) brings together all the known neighbourhoods of x 0 which distance from x 0 is lower than the range of the variogram. The coefficients λ i are calculated by resolving the followed system: Figure 2: kriging system (μ is a Lagrange multiplier) We can easily see that the coefficients mainly depend on the value of the variogram. Separate data Experts know that the behaviour of the contact wire degradation is not the same according to different parts of the wire: connectors can have an impact on the thickness of the wire. In order to be sure that these differences will not impact the model, data are classified depending on their relative position to the connectors. Three types of position are distinguished: - between 0 and 40 cm before the connector - between 0 and 40 cm after the measurement blind zone * induced by the connector - all other positions * As explained in the second part of this article, CYBERNETIX is very sensitive to the location of the connectors. In fact, the system is not able to take measurements of the thickness at the place of

5 connectors, and needs a few seconds to start again taking measures after a connector. This implies no measure at the location of the connector neither along a few centimetres after it. Moreover, other variables have clearly an influence on the degradation of the contact wire. As the classification is not finished, as a first step, we differentiate data according to the traffic the wire is supporting and the year the wire has been set. Of course, before modeling we also separate data according to the type of catenary. All these distinctions lead to consider a lot of different groups of data and to compute many models. Model Here are presented the results of the kriging interpolation on one class of data. This class regroups the sections of contact wire of the Tours-Bordeaux line which present the following characteristics: - wire with a section of 107 mm² - wire with a round geometry - second track - left wire - positions outside the connectors - wire set in traffic About 180 m of line are represented in this group which contains observations. Thanks to these observations, we draw the next variogram: Figure 3: empirical variogram adjusted by a spherical model The empirical variogram (green curve adjusting black points) is adjusted by a theoretical one (blue curve). Actually, the blue curve is a theoretical variogram based on a spherical model. Since the function of the empirical variogram is unknown, the function of the blue curve is that used in the kriging system. On this example, the range of the variogram is about 35 cm. That is to say, two points separated by less than 35 cm have their thicknesses highly correlated. On the contrary, if we consider two points separated by more than 35 cm, the thickness of one point can not give a lot of information on the thickness of the second point. Therefore, in order to estimate the thickness of one point, we will use only the points (which thickness is known) which distances from it are smaller than 35 cm. The values of the variogram are expressed in 10-4 mm² (since the thicknesses are assumed to be known with a 10-2 mm accuracy). Therefore, we can deduce from the graph that the thicknesses have very close values on this section: in fact, the bigger squared difference (at fixed distance) does not exceed mm². Thanks to the adjusted variogram, we can now interpolate the unknown thicknesses with a kriging model.

6 Figure 4: kriging interpolation results First remark: The spaces without points are due to the connectors. In fact, the measurements just before or just after connectors are not taken into account here. This graph reveals quite well the frequency and the regularity of the connectors occurrences. The interpolated thicknesses seem to be rather close to the real measured, especially since the accuracy of the measures is very fine. However, to assess the quality of the kriging interpolation, one needs to observe the errors distribution. Figure 5: errors histogram The errors are calculated by taking the difference between the thickness obtained by kriging interpolation and the real thickness at the same position. The presence of the zero peak is normal: this peak gathers in fact all points which have been used to interpolate (red ones in fig 4). These points are the known points. The kriging interpolation is an exact interpolation, which means that the interpolated value at a known point is equal to the real value. The errors are between mm and 0.10 mm. At worse, there is a tenth of a millimeter difference separating the interpolation and the real value. So, the modeling seems to be quite good.

7 On one hand, the knowledge of the errors distribution can thus be used to obtain an improved evaluation of the kriging quality. On the other hand, it may also be very useful to compute the likelihood not to detect a critical point. Actually, from the errors distribution, we can draw the confidence interval and then evaluate the likelihood to be lower than a critical thickness threshold fixed by the experts. Then this step consisting in fitting the errors distribution is very important, as it contributes to address the main problematic exposed in the introduction. The approach envisaged to fit the distribution is: - exclude (but keep for later calculations) the points causing the zero peak - fit the distribution without the peak At first time, we test if the distribution is normal or not. The distributions already studied can be adjusted by normal laws, but if it is not the case, the alpha stable distributions are contemplated to adjust the kriging errors distribution [4]. Comparison test Due to the lack of information about the physical phenomena of contact wire degradation, we have distinguished as much as possible the data according to very precise criteria. This led us to consider a lot of groups and too many models. But, it is possible that the data of two classes have very similar behaviour. As no physical criterion exists to decide if it is the case or not, we would like to find a statistical criterion to determine if two different classes look very close and enough identical to put them together and compute only one model. So, we could decrease the number of groups of data and at the same time the number of models to compute. Therefore, a statistical test is contemplated Temporal modeling First, we need data to model temporal degradation, which means contact wire thickness measured on the same section for several years. This type of data is given by the so called 'fiches de vie' (life sheets) which result from the manual maintenance. A first study of feasibility has been carried out on the subject: some data have already been collected. These data are not sufficient to describe as precisely as needed the catenary system because they come from a single line. Therefore, statistical analyses are currently being carried out to develop a representative 'training sample' of data. Training sample Thanks to the network classification, we can constitute some classes with homogeneous contact wire lifespan. We would like to get some data to depict every considered class. In order to collect the necessary data, we decided to select for each class about 20 fiches de vie which will be after entered in a database. The 20 selected fiches de vie must represent as well as possible all the considered classes. Therefore, we choose for each class 20 fiches which lifespan histogram is as close as possible to the lifespan histogram from all the class. Moreover, we focus on the fiches which have the following operational qualities: - clearly legible (because the measures are handwritten) - most years of measure - most sections measured Once the sampling done, the fiches selected can be entered as data. This process leads to collect enough data to fit a model for each created class. The following sketch sums up the process:

8 Figure 6: sample process Model In temporal dimension, a first study has shown that the Gamma distribution fits quite well the decrease of thickness between two years. The parameters of the Gamma distribution are, however, not the same according to the observed decrease of thickness. Therefore, a homogenous Gamma distribution can not be used to model the degradation over a long period of time. To overcome this difficulty, a generalized Gamma approach is being considered [5]. 4. Conclusion As we saw, kriging interpolation seems to bring good result to fit the spatial degradation of the catenary contact wire. However, the study of the kriging errors will enable to better appreciate the quality of this model and is also an important step to get the likelihood not to detect a critical point. Nevertheless, some difficulties are encountered when using kriging method. In fact, the kriging interpolation can not be relied on if the variogram is irregular. This may occur for some class near the connectors, for which, kriging can not be used. Other methods, complementary to kriging for these particular classes, are under study to get the likelihood not to detect a critical point. A possible way to explore would be to estimate the density of the thickness at a position conditionally to the thicknesses of the two nearest known points. Temporal modeling will also need further efforts, as soon as a complete set of input data will be available, in order to assess the feasibility of a generalized Gamma approach. As a prospect, bi-dimensional models are contemplated: we would like to incorporate the spatial dimension along with the temporal dimension because of the dependence that seems to exist. Acknowledgements The authors acknowledge the support of the ANRT, the French National Association for Technical Research (CIFRE fellowship 2006/0970).

9 References [1] S. Tufféry, Data Mining et statistique décisionnelle, éditions Technip, 2006 [2] J.-P. Chilès and P. Delfiner, Geostatistics: Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics, 1999 [3] G. Matheron, Estimer et Choisir, Les Cahiers du Centre de Morphologie Mathématique de Fontainebleau, Ecole des Mines de Paris, 1978 [4] L. d Estampes, Traitement statistique des processus alpha-stables, Tests séquentiels tronqués, PhD thesis, 2003 [5] T. Stawart, L. Strijbosch, H. Moors, P. van Batenburg, A Simple Approximation to the Convolution of Gamma Distributions